Insertion-deletion codes (insdel codes for short) play an important role in synchronization error correction. The higher the minimum insdel distance, the more insdel errors the code can correct. Haeupler and Shahrasbi established the Singleton bound for insdel codes: the minimum insdel distance of any $[n,k]$ linear code over $\mathbb{F}_q$ satisfies $d\leq2n-2k+2.$ There have been some constructions of insdel codes through Reed-Solomon codes with high capabilities, but none has come close to this bound. Recently, Do Duc {\it et al.} showed that the minimum insdel distance of any $[n,k]$ Reed-Solomon code is no more than $2n-2k$ if $q$ is large enough compared to the code length $n$; optimal codes that meet the new bound were also constructed explicitly. The contribution of this paper is twofold. We first show that the minimum insdel distance of any $[n,k]$ linear code over $\mathbb{F}_q$ satisfies $d\leq2n-2k$ if $n>k>1$. This result improves and generalizes the previously known results in the literature. We then give a sufficient condition under which the minimum insdel distance of a two-dimensional Reed-Solomon code of length $n$ over $\mathbb{F}_q$ is exactly equal to $2n-4$. As a consequence, we show that the sufficient condition is not hard to achieve; we explicitly construct an infinite family of optimal two-dimensional Reed-Somolom codes meeting the bound.
翻译:在同步错误校正中, 插入运算代码( 短短代码) 具有重要作用 。 在同步错误校正中, 最小值越高, 代码就越能校正。 海普勒 和 Shahrasbi 建立了单吨代码的内嵌代码 : $\ mathbb{F ⁇ qqq$ 上的任何$的线性代码, 最小值为 $d\leqq2n-2k+2. $; 满足新约束的最佳代码的构建也非常明确。 本文的贡献是双倍的。 我们首先显示, 任何 $[ led- Solo 代码的最小值距离, 但没有接近于此约束 。 最近, Do Duc 和 Sharasbi 建立了任何 $ $ $[ nk] Reed- Solomon 代码的内最小值为 $xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx